[Cool] Benford's Law

http://plus.maths.org/issue9/features/benford/

Still blows my mind every time.


So, here's a challenge. Go and look up some numbers. A whole variety of naturally-occuring numbers will do. Try the lengths of some of the world's rivers, or the cost of gas bills in Moldova; try the population sizes in Peruvian provinces, or even the figures in Bill Clinton's tax return. Then, when you have a sample of numbers, look at their first digits (ignoring any leading zeroes). Count how many numbers begin with 1, how many begin with 2, how many begin with 3, and so on - what do you find?

You might expect that there would be roughly the same number of numbers beginning with each different digit: that the proportion of numbers beginning with any given digit would be roughly 1/9. However, in very many cases, you'd be wrong!

Surprisingly, for many kinds of data, the distribution of first digits is highly skewed, with 1 being the most common digit and 9 the least common. In fact, a precise mathematical relationship seems to hold: the expected proportion of numbers beginning with the leading digit n is log10((n+1)/n).

This relationship, shown in the graph of Figure 1 and known as Benford's Law, is becoming more and more useful as we understand it better. But how was it discovered, and why on earth should it be true?

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